Why is cài not needed here?
tờ is the classifier for a sheet of paper, so it also applies to newspapers.
What's the difference between "tờ báo" and simply "báo"?
Your interesting question ! You do not use the word "tờ", when before "báo" had no quantity. Eg: - Ai mua BÁO không ? - Em có bán BÁO ngày hôm qua không ? Bán cho tôi hai TỜ !